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%I #28 May 07 2016 21:43:48
%S 3,19,349,2909,15377,128983,1319411,17797519,94097539,6927837559,
%T 48486712787,968068681519,1472840004019,129001208165719
%N First term of smallest sequence of n consecutive weak primes.
%C Erdős called a weak prime A051635 an "early prime," defined to be one which is less than the arithmetic mean of the prime before it and the prime after it. He conjectured that there are infinitely many consecutive pairs of early primes, and offered $100 for a proof and $25000 for a disproof. See Kuperberg 1992.
%C I make the stronger conjecture that the sequence a(n) is infinite.
%C a(1) = A051635(1), a(2) = A054820(1), a(3) = A054824(1), a(4) = A054829(1), a(5) = A054835(1).
%C a(n) is the prime following A158939(n+1). [Follows from the definitions] - _Chris Boyd_, Mar 28 2015
%H Greg Kuperberg, <a href="http://www.math.niu.edu/~rusin/known-math/93_back/prizes.erd">The Erdos kitty: At least $9050 in prizes!</a>, Newsgroups: rec.puzzles, sci.math, 1992. [Broken link]
%H Greg Kuperberg, <a href="/A051635/a051635.erd.txt">The Erdos kitty: At least $9050 in prizes!</a>, Newsgroups: rec.puzzles, sci.math, 1992. [Cached copy]
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Weak_prime">Weak prime</a>
%F a(n) = min{p(i): 2*p(i+j) < p(i+j-1) + p(i+j+1), j = 0,1,..,n-1}.
%e The primes 19 < (17+23)/2 and 23 < (19+29)/2 are the smallest pair of consecutive weak/early primes, so a(2) = 19.
%Y Cf. A051634, A051635, A054820, A054824, A054829, A054835, A158939.
%K nonn,more
%O 1,1
%A _Jonathan Sondow_, Oct 13 2013
%E a(6) corrected by and a(7)-a(13) from _Giovanni Resta_, Jan 16 2014
%E a(14) from _Giovanni Resta_, Apr 19 2016
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